Finite Section Method Research Articles

Finite sections method for Hessenberg matrices

Let D be a Hessenberg matrix and D the closed operator associated to it. In this work, we study the approximation of the resolvent of D by the resolvents of the truncated matrices D n of D. A concept of determinacy is introduced analogous to the one for Jacobi matrices. We apply our results to some questions in rational approximation.

A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

We consider the numerical treatment of second kind integral equations on the real line of the form $$\phi (s) = \psi (s) + \int_{ - \infty }^{ + \infty } {K(s - t)z(t)} \phi (t)dt, s \in \mathbb{R},$$ (abbreviated? =? +K z ?) in which? ?L 1(?),z ?L ? (?), and? ?BC(?), the space of bounded continuous functions on ?, are assumed known and? ?BC(?) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [?A, A]) via bounds on (I ? K z )?1 as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on ? is then analysed: in the case whenz is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases wherez is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that ifz (related to the boundary impedance in the application) takes values in an appropriate compact subsetQ of the complex plane, then the difference between?(s) and its finite section approximation computed numerically using the iterative scheme proposed is ≤C 1[khlog(1/kh)+(1??)?1/2(kA)?1/2] in the interval [??A, ?A] (?<1), forkh sufficiently small, wherek is the wavenumber andh the grid spacing. Moreover this numerical approximation can be computed in ≤C 2 N logN operations, whereN = 2A/h is the number of degrees of freedom. The values of the constantsC 1 andC 2 depend only on the setQ and not on the wavenumberk or the support ofz.

Four short stories about Toeplitz matrix calculations

The stories told in this paper are dealing with the solution of finite, infinite, and biinfinite Toeplitz-type systems. A crucial role plays the off-diagonal decay behavior of Toeplitz matrices and their inverses. Classical results of Gelfand et al. on commutative Banach algebras yield a general characterization of this decay behavior. We then derive estimates for the approximate solution of (bi)infinite Toeplitz systems by the finite section method, showing that the approximation rate depends only on the decay of the entries of the Toeplitz matrix and its condition number. Furthermore, we give error estimates for the solution of doubly infinite convolution systems by finite circulant systems. Finally, some quantitative results on the construction of preconditioners via circulant embedding are derived, which allow us to provide a theoretical explanation for numerical observations made by some researchers in connection with deconvolution problems.

Band-dominated operators with operator-valued coefficients, their Fredholm properties and finite sections

The central theme of the present paper are band and band-dominated operators, i.e. norm limits of band operators. In the first part, we generalize the results from [24] and [25] concerning the Fredholm properties of band-dominated operators and the applicability of the finite section method to the case of operators with operator-valued coefficients. We characterize these properties in terms of the limit operators of the given band-dominated operator. The main objective of the second part is to apply these results to pseudodifferential operators on cones in IR n which is possible after a suitable discretization.

On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering

On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering

On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering

AbstractWe consider the Dirichlet and Robin boundary value problems for the Helmholtz equation in a non‐locally perturbed half‐plane, modelling time harmonic acoustic scattering of an incident field by, respectively, sound‐soft and impedance infinite rough surfaces.Recently proposed novel boundary integral equation formulations of these problems are discussed. It is usual in practical computations to truncate the infinite rough surface, solving a boundary integral equation on a finite section of the boundary, of length 2A, say. In the case of surfaces of small amplitude and slope we prove the stability and convergence as A→∞ of this approximation procedure. For surfaces of arbitrarily large amplitude and/or surface slope we prove stability and convergence of a modified finite section procedure in which the truncated boundary is ‘flattened’ in finite neighbourhoods of its two endpoints. Copyright © 2001 John Wiley &amp; Sons, Ltd.

Algebras of Approximation Sequences: Finite Sections of Band-Dominated Operators

We develop the stability theory for the finite section method for general band-dominated operators on lp spaces over Zk. The main result says that this method is stable if and only if each member of a whole family of operators – the so-called limit operators of the method – is invertible and if the norms of these inverses are uniformly bounded.

Finite Section Method for Linear Ordinary Differential Equations

Sufficient conditions are given for the solution of a linear ordinary differential equation on the half line to be obtained as the limit of solutions of corresponding equations on finite intervals. Both the time-variant and the time-invariant case are considered, and in the latter case the sufficient conditions referred to above are also shown to be necessary. Included are applications to a differential equation that appears in linear transport theory, and to integral equations with semi-separable kernels.

Spectral approximation of Wiener-Hopf operators with almost periodic generating function

The paper deals with spectral approximation of Wiener-Hopf operators acting on Lp -spaces by their finite sections. The generating functions of the Wiener-Hopf operators are supposed to be continuous plus almost periodic.While the usual spectra of the finite sections drastically fail to converge to the spectrum of the Wiener-Hopf operator,it turns out that other spectral approximants, viz. the pseudospectra and the numerical ranges, do converge perfectly.The proof requires a modified approach to the finite section method for Wiener-Hopf operators. This note generalizes results obtained by Böttcher, Grudsky and Silbermann for the case of continuous generating functions.

Rates of convergence for the approximation of dual shift-invariant systems in ?2(?)

A shift-invariant system is a collection of functions {gm,n} of the form gm,n(k)=gm(k−an). Such systems play an important role in time-frequency analysis and digital signal processing. A principal problem is to find a dual system γm,n(k)=γm(k−an) such that each functionf can be written asf= ∑〈f, γm,n〉gm,n. The mathematical theory usually addresses this problem in infinite dimensions (typically in L2 (ℝ) or l2(ℤ)), whereas numerical methods have to operate with a finite-dimensional model. Exploiting the link between the frame operator and Laurent operators with matrix-valued symbol, we apply the finite section method to show that the dual functions obtained by solving a finite-dimensional problem converge to the dual functions of the original infinite-dimensional problem in l2(ℤ). For compactly supported gm, n (FIR filter banks) we prove an exponential rate of convergence and derive explicit expressions for the involved constants. Further we investigate under which conditions one can replace the discrete model of the finite section method by the periodic discrete model, which is used in many numerical procedures. Again we provide explicit estimates for the speed of convergence. Some remarks on tight frames complete the paper.

Fredholm theory and finite section method for band-dominated operators

The topics of this paper are Fredholm properties and the applicability of the finite section method for band operators onl p -spaces as well as for their norm limits which we call band-dominated operators. The derived criteria will be established in terms of the limit operators of the given band-dominated operator. After presenting the general theory, we present its specifications to concrete classes of band-dominated operators.

Numerical ranges of large Toeplitz matrices

A Bananch algebraic approach is proposed to study the asymptotic behaviour of the numerical ranges of certain (finite) approximation matrices of (infinite) operators. The approach works for large classes of approximation methods; it is examined in detail here for the finite sections of Toeplitz operators and of operators which are generated by Toeplitz operators. The basic ingredient is a precise knowledge of the finite section method for Toeplitz and related operators.

Finite Section Method in some Algebras of Multiplication and Convolution Operators and a Flip

This paper is concerned with the applicability of the finite section method to operators belonging to the closed subalgebra of \mathcal L(L^2(\mathbb R)) generated by operators of multiplication by piecewise continuous functions in \mathbb R , convolution operators - also with piecewise continuous generating functions - and the flip operator (Ju)(x) = u(–x) . For this, a larger algebra of sequences is introduced, which contains the special sequences we are interested in. There is a direct relationship between the applicability of the finite section method for a given operator and the invertibility of the corresponding sequence in this algebra. Exploring this relationship, the methods of essentialization, localization and identification of the local algebras through construction of locally equivalent representations are used and so useful invertibility criteria are derived. Finally, examples are presented, including explicit conditions for the applicability of the finite section method to a Wiener-Hopf plus Hankel operator with piecewise continuous symbols, and some relations between the approximation operators and the limit operator are discussed.

Infinite Toeplitz and Hankel Matrices with Operator-Valued Entries

Infinite Toeplitz matrices with operator-valued entries arise, for example, when interpreting Wiener–Hopf integral operators on $L^2 (0,\infty )$ as matrices acting on the direct sum of countably many copies of $L^2 (0,1)$ . This paper concerns the question of asymptotically inverting such infinite Toeplitz matrices by having recourse to their finite principal sections. As expected from the corresponding theories for the scalar and matrix-valued cases, this problem leads to the investigation of compactness properties of infinite Hankel matrices. By introducing the concept of $Q_n $-compact operators on spaces of square-summable sequences with values in a separable Hilbert space, criteria for the applicability of the finite section method to Toeplitz operators with symbols in $C + H^\infty $, in $PC$, or with locally sectorial symbols are established.

Asymptotic Moore-Penrose invertibility of singular integral operators

The topic of the paper is the asymptotic Moore-Penrose inversion of singular integral operators by means of the trigonometric collocation and the finite section method. Two versions (a “weak” and a “strong” one) of these methods are considered. Necessary and sufficient conditions for the applicability of the weak method, and sufficient conditions for the strong method, are derived. These conditions involve a certain asymptotic resp. exact kernel structure of the approximation matrices.

Finite sections for singular integral operators by weighted Chebyshev polynomials

A finite section method for the approximate solution of singular integral equations with piecewise continuous coefficients on intervals is considered. The problem is transformed in such a way that results which were previously obtained for singular integral equations on the unit circle using localization methods in Banach algebras are applicable to it. Thus, necessary and sufficient conditions for the stability of the approximation method can be proved.

The finite section method for Moore-Penrose inversion of Toeplitz operators

It is shown that for certain classes of infinite block Toeplitz matricesT(a)=[a j-k ] 0 ∞ the Moore-Penrose inverses of the finite sectionT n (a)=[a j-k ] 0 n−1 converge to the Moore-Penrose inverse ofT(a). Furthermore the convergence for modified finite section methods and the finite section method for Wiener-Hopf integral and related operators are studied.

Spline Approximation Methods Cutting Off Singularities

The topic of this paper is some types of singular behavior of spline approximation methods for one-dimensional singular integral operators which are caused by discontinuities in the coefficients or by non-smooth geometries of the underlying curves. These singularities can be cutted off by a modification of the approximation method which is closely related to the finite or infinite section method for discrete Toeplitz operators. Using Banach algebra techniques, one can derive stability criteria for a large variety of modified methods (including Galerkin, collocation, and qualocation.

On Asymptotic Behavior at Infinity and the Finite Section Method for Integral Equations on the Half-Line

e consider integral equations on the half-line of the form and the finite section approximation to x obtained by replacing the infinite limit of integration by the finite limit β. We establish conditions under which, if the finite section method is stable for the original integral equation (i.e. exists and is uniformly bounded in the space of bounded continuous functions for all sufficiently large β), then it is stable also for a perturbed equation in which the kernel k is replaced by k + h. The class of perturbations allowed includes all compact and some non-compact perturbations of the integral operator. Using this result we study the stability and convergence of the finite section method in the space of continuous functions x for which ()()()=−∫∞dttxt,sk)s(x0()syβxβx()sxsp+1 is bounded. With the additional assumption that ()(tskt,sk−≤ where ()()(),qsomefor,sassOskandRLkq11>+∞→=∈− we show that the finite-section method is stable in the weighted space for ,qp≤≤0 provided it is stable on the space of bounded continuous functions. With these results we establish error bounds in weighted spaces for x - xβ and precise information on the asymptotic behaviour at infinity of x. We consider in particular the case when the integral operator is a perturbation of a Wiener-Hopf operator and illustrate this case with a Wiener-Hopf integral equation arising in acoustics.

Finite Section Method for the Double Layer Potential Operator over Polyhedral Boundaries

AbstractIn this paper we consider the finite section method for the solution of the double layer potential equation corresponding to Laplace's equation in a three‐dimensional polyhedron. We prove the stability of our method in case of special polyhedrons.